ECDH Key Exchange

The Same Idea, Different Math

ECDH is Diffie-Hellman, but using elliptic curves instead of modular exponentiation.

The goal is the same: create a shared secret over a public channel.

The Setup

Alice and Bob agree on:

• An elliptic curve
• A generator point $G$

These are public. Everyone knows them.

The Protocol

Alice’s side:

1. Pick a secret number $a$ (her private key)
2. Compute $A = aG$ (her public key)
3. Send $A$ to Bob

Bob’s side:

1. Pick a secret number $b$ (his private key)
2. Compute $B = bG$ (his public key)
3. Send $B$ to Alice

Creating the Shared Secret

Alice computes:

$a \times B = a \times (bG) = abG$

Bob computes:

$b \times A = b \times (aG) = abG$

Both get the same point $abG$.

Why It Works

Scalar multiplication is commutative:

$a(bG) = b(aG) = (ab)G$

The order doesn’t matter. Both paths lead to the same point.